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58 III METRIC SPACES

("open covering") there exists a finite subfamily (U_A)_AeH (H c L and finite)
which is a covering ofE.

A metric space E is called precompact if it satisfies the following condi-
tion: for any s > 0, there is a finite covering ofE by sets of diameter < s. This
is immediately equivalent to the following property: for any s > 0, there is a
finite subset ¥ofE such that d(x, F) < e/or every xeE.

In the theory of metric spaces, these notions are a substitute for the notion
of "finiteness" in pure set theory; they express that the metric space is, so
to speak, " approximately finite." Note that, from the definition, it follows
that compactness is a topological notion, but precompactness is not (see
remark after (3.17.6)),

(3.16.1) For a metric space E, the following three conditions are equivalent:

(a) E is compact;

(b) any infinite sequence in E has at least a cluster value;

(a) =>(b): Let (x_n) be an infinite sequence in the compact space E, and
let F_rt be the closure of the set {x_n, x_n+l,..., x_n+p,...}. We prove there is a
point belonging to all F_n. Otherwise, the open sets U_M = E - F_w would form a
covering of E, hence there would exist a finite number of them, U_nj5..., U_rtkforming a covering of E; this would mean that F_ni n F_n2 n • - • n F_nk = 0;
but this is absurd, since if n is greater than m&x(n_iy..., %), F_n (which is not
empty by definition) is contained in all the F_nj (1 ^ / < k). Hence the inter-

section f) F_n contains at least a point a. By (3.13.11) and the definition of a

n=l

cluster point, a is a cluster value of (x_n).

(b)=>(c): First any Cauchy sequence in E has a cluster value, hence is
convergent by (3.14.2), and therefore E is complete. Suppose E were not
precompact, i.e. there exists a number a > 0 such that E has no finite covering
by balls of radius oc. Then we can define by induction a sequence (X_M) in the
following way: x₁ is an arbitrary point of E; supposing that
d(x_i9 Xj) > a for i ^/, 1 < i < /i - 1, 1 < y < w •- 1, the union of the balls of
center x_t (1 < i ^ n - 1) and radius a is not the whole space, hence there is
x_n such that d(x_t ₉x_n)^a for z < n. The sequence (x_n) cannot have a cluster
value, for if a were such a value, there would be a subsequence (x_nk) converging
to a, hence we would have d(a, x_ni) < a/2 for k ^ k_Q, and therefore
^rf(*»h^^xn_k}<^foTh^k_Q,k^kQ,h^k_} contrary to the definition of (*„).

(c) => (a): Suppose we have an open covering (U_A)_A _e _L of E such that no
finite subfamily is a covering ofE. We define by induction a sequence (B_w) of



	58 III METRIC SPACES ("open covering") there exists a finite subfamily (U_A)_AeH (H c L and finite) which is a covering ofE. A metric space E is called precompact if it satisfies the following condi- tion: for any s > 0, there is a finite covering ofE by sets of diameter < s. This is immediately equivalent to the following property: for any s > 0, there is a finite subset ¥ofE such that d(x, F) < e/or every xeE. In the theory of metric spaces, these notions are a substitute for the notion of "finiteness" in pure set theory; they express that the metric space is, so to speak, " approximately finite." Note that, from the definition, it follows that compactness is a topological notion, but precompactness is not (see remark after (3.17.6)),

	(3.16.1) For a metric space E, the following three conditions are equivalent: (a) E is compact; (b) any infinite sequence in E has at least a cluster value; (c) E fs precompact and complete. (a) =>(b): Let (x_n) be an infinite sequence in the compact space E, and let F_rt be the closure of the set {x_n, x_n+l,..., x_n+p,...}. We prove there is a point belonging to all F_n. Otherwise, the open sets U_M = E - F_w would form a covering of E, hence there would exist a finite number of them, U_nj5..., U_rtkforming a covering of E; this would mean that F_ni n F_n2 n • - • n F_nk = 0; but this is absurd, since if n is greater than m&x(n_iy..., %), F_n (which is not empty by definition) is contained in all the F_nj (1 ^ / < k). Hence the inter- 00 section f) F_n contains at least a point a. By (3.13.11) and the definition of a n=l cluster point, a is a cluster value of (x_n). (b)=>(c): First any Cauchy sequence in E has a cluster value, hence is convergent by (3.14.2), and therefore E is complete. Suppose E were not precompact, i.e. there exists a number a > 0 such that E has no finite covering by balls of radius oc. Then we can define by induction a sequence (X_M) in the following way: x₁ is an arbitrary point of E; supposing that d(x_i9 Xj) > a for i ^/, 1 < i < /i - 1, 1 < y < w •- 1, the union of the balls of center x_t (1 < i ^ n - 1) and radius a is not the whole space, hence there is x_n such that d(x_t ₉x_n)^a for z < n. The sequence (x_n) cannot have a cluster value, for if a were such a value, there would be a subsequence (x_nk) converging to a, hence we would have d(a, x_ni) < a/2 for k ^ k_Q, and therefore ^rf(»h^^xn_k}<^foTh^k_Q,k^kQ,h^k_} contrary to the definition of („). (c) => (a): Suppose we have an open covering (U_A)_A _e _L of E such that no finite subfamily is a covering ofE. We define by induction a sequence (B_w) of